Monday, May 29, 2006

Why the bus comes just when you are about to give up on it

Explanation 1: The wait time for the bus follows a normal distribution with a mean of t and some variance. You have some sense of the mean, and perhaps the variance. Once you have waited that mean amount of time, you start to get impatient and want to give up on the bus arriving. However, the probability of the bus arriving at t+1 is much greater than the probability of it arriving at time t+2 and etc. So as you wait past time t, the probability of the bus not showing up very soon becomes vanishingly unlikely.

Redoing this with, say, an exponential process would yield very different results.

Explanation 2: Once you are about to give up, either the bus comes and the bus has arrived just when you are about to give up, or else you do give up and you walk or take a taxi or something. Point being, that it is definitionally true that the bus arrives just when you are about to give up, because otherwise you give up and you never find out how long it takes the bus to arrive.

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